# Hexahedron

The word “hexahedron” is derived from the the two Greek words “hexa”, which means six, and “hedra”, which means area, or surface. That leads directly to the specified object: a geometrical body consisting of six conjoined areas.

It shows an enormously high symmetry, since it consists of six equal areas, each corresponding to a square. That square himself is again a shape of highest symmetry (4 symmetry axis, point symmetry, axially symmetrical to multiples of 90°).

The high symmetry of the hexahedron is for example visible by the fact, that it remains always exactly the same body, independent of the area on which it is positioned. Furthermore it has point symmetry. Besides that it is for sure the most popular one of the Platonic bodies, since it is a really widespread shape, also known as cube, or dice.

Looking closer to the hexahedron reveals the following, geometrical characteristics – it has 8 vertices, 12 edges, and as well 3 edges as 3 areas joined to a vertex.

The following parameters are effective:

• the height, when positioned on an area, comes up to the length of an edge
• the surface area comes up to six times the square of the length of an edge
• the volume comes up to the cube of the length of an edge
• all angles are corresponding to the right angle, so 90°

The one who wants to deal more profoundly with the geometrical characteristics of an hexahedron will find a lot of material quite suitable for self studies on Internet. There can be found a lot on topics as in-circle, edge-circle, circumcircle, symmetry axis, symmetry areas, measures, and so on.

But the direct experimentation on the object itself is more valuable in my point of view, to really grasp and conceive in detail what you are investigating – whatever the concrete manner for that may be. The resulting perceptions are not so formal and theoretical, more apparent, and may be directly experienced in a playful examination of the object.

So you have for example always a body that is dual to a Platonic body, resulting again in a Platonic body. Through a geometrical transformation a Platonic body is converted into another one, thus revealing a kind of relatedness among them. In case of the hexahedron his dual partner is the octahedron. A short plausibility check gives rise to that relation: it has six areas, so that the individual center points will result in six vertices, which is only the case for the octahedron within the Platonic bodies.

## Practical Example

Another way of experimenting with this shape is the tesselation, i.e. the identification of a way how to consistently fill space with a Platonic body. How does it look like for the hexahedron – is it feasible? One possibility of determination is the construction of a hexahedron of twice the size, or duplicated length of an edge, out of hexahedrons of original size. But imagination lets you already assume, that this procedure will be easy, as can also be seen in the following pictures…

First step: an hexahedron.

Next step: two hexahedrons.

Next step: three hexahedrons.

Next step: four hexahedrons.

Next step: five hexahedrons.

Next step: six hexahedrons.

Next step: seven hexahedrons.

Last step: eight hexahedrons.

For the construction of an hexahedron of twice the length of an edge eight hexahedrons of original size are required to proceed the tesselation, which is just like that feasible.